Buoyancy-gradient mixing-length closure ($K\sim d^2\sqrt{G_z}$)

For positive dense-fluid reduced gravity $G$ and upward coordinate $z$, an unstable gradient $G_z>0$ supplies buoyant work of order $d^2{G}_z$ over mixing length $d$. velocity $u\sim d\sqrt{G_z}$ and eddy diffusivity $K\sim ud$ follow. Order-one constants depend on the turbulence model. This closure is for an unstable gradient; its square root must not be applied to a stable negative gradient.

An upward volume flux $Q$ and a downward turbulent buoyancy transport $d^4(G_z{)}^{3/2}$ balance in a steady reactor. Matching $G=G_z=0$ below $-H$ gives $QG=d^4(G_z{)}^{3/2}$. Integrating yields the cubic profile above the front. If downward source buoyancy flux is $B_s$, then $H=3d^{8/3}{B}_s^{1/3}/Q$. The corresponding eddy diffusivity is $K=Q(z+H)/(3d^2)$. All these prefactors use the same area and mixing-length normalization; an actual cross-sectional area $S$ replaces the denominator in $K$ by $3S$.

In the area normalization $d^2$, the buoyancy-gradient mixing-length closure transports dense-fluid buoyancy downward with magnitude $\mathcal{F}=d^{2}K{G}_z=d^4(G_z{)}^{3/2}$. Its convergence in an upward coordinate is ${\mathcal{F}}_z$. Transport has units $L^4{T}^{-3}$ and convergence units $L^3{T}^{-3}$. Constant positive $G_z$ has nonzero transport and zero convergence; the two quantities cannot be identified.